Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set $A$ we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s...

We show here that a wide class of integral inequalities concerning functions on $[0,1]$ can be obtained by purely combinatorial methods. More precisely, we obtain modulus of continuity or other high order norm estimates for functions satisfying conditions of the type ${\int }_0^1{\int }_0^1\Psi \left(\frac{f\left(x\right)-f\left(y\right)}{p(x-y)}\right)dxdy\<\infty$ where $\Psi \left(u\right)$ and $p\left(u\right)$ are monotone increasing functions of $\left|u\right|$.Several applications are also derived. In particular these methods are shown to yield a new condition for path continuity of general stochastic processes

In this article we demonstrate basic properties of the continuous functions from R to Rn which correspond to state space equations in control engineering.

Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set ${\Phi }^{-1}\left(E\right)$ is residual whenever E is residual in...